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TWEET # demonio a guest Nov 19th, 2019 85 Never
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1. % [H,Q] = hessred(A)
2. %
3. % Compute the Hessenberg decomposition H = Q’*A*Q using
4. % Householder transformations.
5. %
6. function [H,Q] = hessred(A)
7.
8. n = length(A);
9. Q = eye(n); % Orthogonal transform so far
10. H = A; % Transformed matrix so far
11.
12. for j = 1:n-2
13.
14. % -- Find W = I-2vv’ to put zeros below H(j+1,j)
15. u = H(j+1:end,j);
16. u(1) = u(1) + sign(u(1))*norm(u);
17. v = u/norm(u);
18.
19. % -- H := WHW’, Q := QW
20. H(j+1:end,:) = H(j+1:end,:)-2*v*(v’*H(j+1:end,:));
21. H(:,j+1:end) = H(:,j+1:end)-(H(:,j+1:end)*(2*v))*v’;
22. Q(:,j+1:end) = Q(:,j+1:end)-(Q(:,j+1:end)*(2*v))*v’;
23.
24. end
25.
26. end
27. % [H] = hessqr_basic(H)
28. %
29. % Compute one basic (unshifted) implicit Hessenberg QR step via
30. % Householder transformations.
31. %
32. function H = hessqr_basic(H)
33.
34. n = length(H);
35. V = zeros(2,n-1);
36.
37. % Compute the QR factorization
38. for j = 1:n-1
39.
40. % -- Find W_j = I-2vv’ to put zero into H(j+1,j)
41. u = H(j:j+1,j);
42. u(1) = u(1) + sign(u(1))*norm(u);
43. v = u/norm(u);
44. V(:,j) = v;
45.
46. % -- H := W_j H
47. H(j:j+1,:) = H(j:j+1,:)-2*v*(v’*H(j:j+1,:));
48.
49. end
50.
51. % Compute RQ
52. for j = 1:n-1
53.
54. % -- H := WHW’, Q := QW
55. v = V(:,j);
56. H(:,j:j+1) = H(:,j:j+1)-(H(:,j:j+1)*(2*v))*v’;
57.
58. end
59.
60. end
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